- how can we study their structures ? - many ways, for example x-ray
diffraction.

Crystallinity - have a repeating unit = unit cell

To define repeating unit use concept of a lattice

A lattice is "an infinite 1,2, or 3-D regular arrangement of points,
each of which has identical surroundings".

Any periodic pattern can be described by placing lattice points at equivalent
positions within each unit of the pattern.

To recover original pattern we add the motif to each lattice point.
(return to top)

1-D lattices. The regular pattern
of wagons below can be described by placing a lattice point at the same
place in each wagon. The arrangement of dots is the lattice, which
has a given repeat distance. The motif is the wagon. The pattern
is recovered by stamping the motif on each lattice point.

Consider each of the patterns below - what is the lattice and unit cell
?
Place lattice points at equivalent positions in the pattern, find smallest
repeat unit that by translation, can cover all space.
All of these patterns have the same Planar Lattice.(square),
but each has a different motif.

Crystal structures repeat in 3 dimensions. The motif can be single
atoms or groups of atoms. Again we assign lattice points to the atomic
structure and produce a Space Lattice.

Space lattice + motif = Crystal Structure

There are 7 unique unit-cell shapes that can fill all 3-D space.
These are the 7 Crystal systems.

We define the size of the unit cell using lattice parameters
(sometimes called lattice constants, or cell parameters). These are
3 vectors, a, b, c. The angles between these vectors are given by
a (angle between b and c), b
(angle between a and c), and g (angle between
a and b).

The Seven Crystal Systems

Although there are only 7 crystal systems or shapes, there are 14
different crystal lattices, called Bravais Lattices. (3
different cubic types, 2 different tetragonal types, 4 different orthorhombic
types, 2 different monoclinic types, 1 rhombohedral, 1 hexagonal, 1 triclinic).
See below.

Real crystals always possess one of these lattice types, but different
crystalline compounds that have the same lattice can have different motifs
and different lattice parameters (these depend upon the chemical formula
and the sizes of the atoms in the unit cell). We will only concern
ourselves with the cubic lattices, though we will refer to the hexagonal
lattice in passing.

Now that we know how to describe crystalline solids, let us examine
different types according to the nature of their bonding. We will
begin with metallic
solids, followed by ionic
solids, and extended
covalentor framework solids.